On the gamma-vector of symmetric edge polytopes

We study $\gamma$-vectors associated with $h^{\ast }$-vectors of symmetric edge polytopes both from a deterministic and a probabilistic point of view. On the deterministic side, we prove nonnegativity of ${\gamma }_2$ for any graph and completely characterize the case when ${\gamma }_2=0$. The latter also confirms a conjecture by Lutz and Nevo in the realm of symmetric edge polytopes. On the probabilistic side, we show that the $\gamma$-vectors of symmetric edge polytopes of most Erd\H{o}s-R'enyi random graphs are asymptotically almost surely nonnegative up to any fixed entry. This proves that Gal’s conjecture holds asymptotically almost surely for arbitrary unimodular triangulations in this setting.